Compact finite differences have been already discussed in this post and following responses: http://lists.mcs.anl.gov/pipermail/petsc-users/2011-November/011006.html.
If I understand well, the matrix for a 2-D (or 3-D) *implicit* solver that makes use of compact FD for the calculation of derivatives is not "banded" because of the ordering of the x-y-z variables in the solution vector destroys any structure, and the matrix is simply sparse (bandwith=sqrt(n)). In that post is actually suggested that a sparse dense solver could be more attractive because of that. However, if the PDE's are solved explicitly in time, the derivatives in all directions may be calculated independently by using an efficient sequential algorithm (o(n)). I am not familiar with any parallel implementations of the banded diagonal algorithm solvers, but I was wondering whether this could be efficiently implemented in Petsc with the existing MA modules. But at that point, I don't know whether there will be any advantage in possibly using an iterative solver that converges to the exact solution in N iterations for a well behaved matrix as the one deriving from compact FD. --g
